To solve the equation $z^2 - i = 0$, we can follow these steps:

Step 1: Rearrange the equation.

$z^2 = i$

Step 2: Express $z$ in Cartesian form.

Let $z = x + iy$, where $x$ and $y$ are real numbers, and $i$ is the imaginary unit.

Thus, the equation becomes: $(x + iy)^2 = i$

Step 3: Expand the left-hand side.

$(x + iy)^2 = x^2 + 2ixy + i^2y^2$ Since $i^2 = -1$, we get: $x^2 - y^2 + 2ixy = i$

Step 4: Separate real and imaginary parts.

The equation $x^2 - y^2 + 2ixy = i$ can be split into its real and imaginary components:

• Real part: $x^2 - y^2 = 0$
• Imaginary part: $2xy = 1$

Step 5: Solve the system of equations.

From the real part:

$x^2 - y^2 = 0 \implies x^2 = y^2 \implies x = \pm y$

From the imaginary part:

$2xy = 1 \implies xy = \frac{1}{2}$

Step 6: Solve for $x$ and $y$.

1. Case 1: $x = y$

   Substituting $x = y$ into $xy = \frac{1}{2}$, we get: $x^2 = \frac{1}{2} \implies x = \pm \frac{1}{\sqrt{2}}$ So, $y = \pm \frac{1}{\sqrt{2}}$.

   Thus, one solution is $z = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}$.

2. Case 2: $x = -y$

   Substituting $x = -y$ into $xy = \frac{1}{2}$, we get: $-x^2 = \frac{1}{2} \implies x^2 = -\frac{1}{2}$ This results in no real solutions, since $x^2 \geq 0$ for all real $x$.

Step 7: Final solutions.

The only valid solution is: $z = \pm \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}$

So, the solutions to the equation $z^2 - i = 0$ in Cartesian form are: $z = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} \quad \text{or} \quad z = -\frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}}$

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Would you like more details on any part of the solution, or do you have any questions about it? Here are some related questions for further practice:

1. How can you solve similar equations involving complex numbers in polar form?
2. How would you verify if $z = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}$ is indeed a solution of the original equation?
3. What happens if the equation were $z^2 + i = 0$ instead of $z^2 - i = 0$?
4. How would you solve a quadratic equation in the complex plane generally?
5. What is the geometric interpretation of squaring a complex number?

Tip: When solving complex equations, separating the real and imaginary parts often simplifies the process!